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13
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 205 (15 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone
, 2011
"... With the help of the GaussLaplace transform for the exit time from a cone of planar Brownian motion, we obtain some infinite divisibility properties for the reciprocal of this exit time. ..."
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Cited by 2 (1 self)
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With the help of the GaussLaplace transform for the exit time from a cone of planar Brownian motion, we obtain some infinite divisibility properties for the reciprocal of this exit time.
Integrability properties and Limit Theorems for the exit time from a cone of planar Brownian motion
, 2011
"... We obtain some integrability properties and some limit Theorems for the exit time from a cone of a planar Brownian motion, and we check that our computations are correct via Bougerol’s identity. ..."
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Cited by 1 (1 self)
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We obtain some integrability properties and some limit Theorems for the exit time from a cone of a planar Brownian motion, and we check that our computations are correct via Bougerol’s identity.
reality: Hybrid Brownian motion with price
, 2009
"... A model of returns for the postcreditcrunch ..."
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A Model of Returns for the PostCreditCrunch environment
, 2009
"... The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival ..."
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The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival model with variable size orders. With simplifications we are led to an SDE mixing both arithmetic and geometric Brownian motions, whose solution is an given by a class of integrals of exponentials of Brownian motions, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of “Pearson diffusion”, or equivalently a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under meanreverting circumstances we arrive naturally at an equilibrium fattailed return distribution with a Student or Pearson Type IV form. Under less restrictive assumptions richer dynamics are possible. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that “25σ ” events are realistic. We exhibit simple example solutions of the FokkerPlanck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviours. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity.
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, 1995
"... The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a ..."
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The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.